| L(s) = 1 | + (−2.47 + 1.34i)3-s + (−0.998 + 2.00i)5-s + (0.191 + 0.255i)7-s + (2.66 − 4.14i)9-s + (−4.15 + 1.89i)11-s + (−4.13 − 3.09i)13-s + (−0.231 − 6.29i)15-s + (−0.227 − 3.17i)17-s + (3.00 + 3.46i)19-s + (−0.816 − 0.373i)21-s + (−3.40 + 3.37i)23-s + (−3.00 − 3.99i)25-s + (−0.386 + 5.40i)27-s + (5.27 + 4.57i)29-s + (1.08 − 0.317i)31-s + ⋯ |
| L(s) = 1 | + (−1.42 + 0.778i)3-s + (−0.446 + 0.894i)5-s + (0.0722 + 0.0965i)7-s + (0.887 − 1.38i)9-s + (−1.25 + 0.571i)11-s + (−1.14 − 0.857i)13-s + (−0.0598 − 1.62i)15-s + (−0.0550 − 0.770i)17-s + (0.688 + 0.794i)19-s + (−0.178 − 0.0814i)21-s + (−0.710 + 0.703i)23-s + (−0.601 − 0.799i)25-s + (−0.0744 + 1.04i)27-s + (0.979 + 0.849i)29-s + (0.194 − 0.0570i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.600 + 0.799i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.600 + 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.225533 - 0.112680i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.225533 - 0.112680i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.998 - 2.00i)T \) |
| 23 | \( 1 + (3.40 - 3.37i)T \) |
| good | 3 | \( 1 + (2.47 - 1.34i)T + (1.62 - 2.52i)T^{2} \) |
| 7 | \( 1 + (-0.191 - 0.255i)T + (-1.97 + 6.71i)T^{2} \) |
| 11 | \( 1 + (4.15 - 1.89i)T + (7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (4.13 + 3.09i)T + (3.66 + 12.4i)T^{2} \) |
| 17 | \( 1 + (0.227 + 3.17i)T + (-16.8 + 2.41i)T^{2} \) |
| 19 | \( 1 + (-3.00 - 3.46i)T + (-2.70 + 18.8i)T^{2} \) |
| 29 | \( 1 + (-5.27 - 4.57i)T + (4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (-1.08 + 0.317i)T + (26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 + (-4.74 + 1.03i)T + (33.6 - 15.3i)T^{2} \) |
| 41 | \( 1 + (0.580 - 0.373i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (1.90 + 3.49i)T + (-23.2 + 36.1i)T^{2} \) |
| 47 | \( 1 + (3.43 + 3.43i)T + 47iT^{2} \) |
| 53 | \( 1 + (-4.05 + 3.03i)T + (14.9 - 50.8i)T^{2} \) |
| 59 | \( 1 + (-2.01 + 0.289i)T + (56.6 - 16.6i)T^{2} \) |
| 61 | \( 1 + (1.29 + 4.41i)T + (-51.3 + 32.9i)T^{2} \) |
| 67 | \( 1 + (0.262 + 0.705i)T + (-50.6 + 43.8i)T^{2} \) |
| 71 | \( 1 + (-3.29 + 7.21i)T + (-46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (-7.11 - 0.509i)T + (72.2 + 10.3i)T^{2} \) |
| 79 | \( 1 + (-0.450 - 3.13i)T + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (0.313 + 1.44i)T + (-75.4 + 34.4i)T^{2} \) |
| 89 | \( 1 + (-11.8 - 3.47i)T + (74.8 + 48.1i)T^{2} \) |
| 97 | \( 1 + (-3.86 + 17.7i)T + (-88.2 - 40.2i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.21443381384645618800307293783, −9.660698607099840500320934118895, −7.987006026545458473356523446738, −7.37852243055030753306469054568, −6.43065915883611388761922170971, −5.28298907936789707574299444723, −5.02300590892358547211600013830, −3.70750396382392009031129639996, −2.56727089810204090126009897561, −0.18020434558896256825010964067,
0.966350379013932375681914251101, 2.47014687208790140011821784812, 4.38159963615895364182306751285, 5.00057856812417240142837125955, 5.84244508981809495306296549061, 6.71320754996148657098113241222, 7.67428795589439583593942039463, 8.226076059105685637418362539837, 9.457139076921357400107851809261, 10.41877454096656256172049884390
